PSFC/JA-10-42
A unified treatment of kinetic effects
in a tokamak pedestal
Peter J. Catto, Gregory Kagan1,
Matt Landreman, and Istvan Pusztai2
1
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Nuclear Engineering, Applied Physics, Chalmers University of Technology and
Euratom-VR Association, SE-41296 Goteborg, Sweden
2
November 2010
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge MA 02139 USA
This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER54109. Reproduction, translation, publication, use and disposal, in whole or in part, by or
for the United States government is permitted.
Submitted to Plasma Phys. Control. Fusion (August 2010)
A unified treatment of kinetic effects in a tokamak pedestal
Peter J. Catto1, Grigory Kagan2, Matt Landreman1, Istvan Pusztai3
1) Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA
2) Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3) Nuclear Engineering, Applied Physics, Chalmers University of Technology and EuratomVR Association, SE-41296 Goteborg, Sweden
e-mail contact: catto@psfc.mit.edu
Abstract
We consider the effects of a finite pedestal radial electric field on ion orbits using a unified
approach. We then employ these modified orbit results to retain finite E×B drift departures
from flux surfaces in an improved drift kinetic equation. The procedure allows us to make
clear distinction between transit averages and flux surface averages when solving this kinetic
equation. The technique outlined here is intended to clarify and unify recent evaluations of
the banana regime decrease and plateau regime alterations in the ion heat diffusivity, the
reduction and possible reversal of the poloidal flow in the banana regime and its
augmentation in the plateau regime, the increase in the bootstrap current, and the
enhancement of the residual zonal flow regulation of turbulence.
1. Introduction
It is standard in tokamak kinetic theory to assume the poloidal ion gyroradius is small
compared to all macroscopic scale-lengths, but in the pedestal this approximation can break
down. This challenge is addressed by our recently developed techniques that account for the
presence of short radial scale lengths in a subsonic pedestal. Using these techniques, we have
calculated both the banana [1,2] and plateau [3] regime modifications to the neoclassical ion
heat flux, ion and impurity flows, the bootstrap current; and extended the residual zonal flow
calculation to the pedestal case [4,5]. Here, we present a general formalism providing deeper
insight into these detailed calculations by considering the effects of the finite pedestal radial
electric field E on the ion orbits by an improved procedure. This unified approach highlights
intricacies of the previous investigations, thereby forming a firm basis for how these kinetic
calculations are best performed. In particular, we present a general method of treating finite
electric field effects on ion orbits that allows us to solve a reduced kinetic equation for the
pedestal modifications to the results of the usual evaluations of neoclassical transport in the
banana and plateau regimes [6-8], and the residual zonal flow calculation [9,10]. The
emphasis herein is on the steps that differ from the usual neoclassical and residual zonal flow
treatments. The basis of the technique is the convenience of employing the canonical angular
momentum as the radial gyrokinetic variable [11]. This choice allows us to perform
calculations in the pedestal by accounting for the presence of strong radial density (and
electron temperature) gradient scale length on the order of a poloidal ion gyroradius ρpi =
ρiB/Bp, where B/Bp >> 1 and ρ i = vi/Ωi is the ion gyroradius with vi = (2Ti/M)1/2 the ion
thermal speed and Ωi = ZeB/Mc the ion gyrofrequency for ions of mass M, charge number Z,
and temperature Ti.
In a subsonic pedestal the E×B drift and the ion diamagnetic drift must cancel to
lowest order [11-13]. In the helium discharges on DIII-D this behavior is verified since the
background ion temperature can be measured directly [14]. In this situation the ions are
electrostatically confined to lowest order and the associated radial electric field is so large
that the E×B drift velocity can compete with the poloidal component of the parallel streaming
to modify trajectories. These modifications introduce electric field dependence into the
neoclassical ion and impurity flows via the usual ion temperature gradient term [1,2]. The ion
temperature pedestal is always at least B/Bp wider than ρpi because of the constraint that the
entropy production must vanish in a banana or plateau regime pedestal [11]. In addition, the
ion heat flux [1,3] and the bootstrap current [2,3] are altered, along with the residual zonal
flow [4,5] response of Rosenbluth and Hinton [9]. In spite of these changes the plasma
remains intrinsically ambipolar [15,16] in the more general sense that E (or more precisely,
the difference between the E×B and ion diamagnetic drifts) remains undetermined until
conservation of toroidal angular momentum is solved, but unlike the core, the heat flux does
depend on the radial electric field as does the ambipolar particle flux.
In section 2 we present a unified approach to retaining the effect of the radial electric
field on the ion orbits to be used for both the banana and plateau regime cases. A drift kinetic
equation valid for treating neoclassical transport in both regimes of collisionality that also
retains the zonal flow residual is efficiently derived in section 3. The generality of the results
in these two sections then allows us to conveniently derive all earlier neoclassical banana
[1,2] and plateau [3] results in section 4, as well as all previous results for the radial electric
field modified zonal flow residual [4] and its orbit squeezing generalization [5] in section 5.
Moreover, the treatment of collisions is streamlined so that the transit averaged collisional
constraints are performed retaining finite drift orbit effects.
€
2. Radial electric field effects on the treatment of ion orbits
When the radial density scale length of the pedestal becomes comparable to the
poloidal ion gyroradius, ρpi, the ion flow speed can only be subsonic if the E × B and ion
diamagnetic flows cancel to lowest order. The lowest order cancellation means that the ions
are electrostatically confined with the radial electric field satisfying (Ze/Ti)dΦ/dr ≈ -dlnpi/dr
€ field results in an
~ 1/ ρ pi >> 1/a, with a the minor radius [11]. Such a strong radial electric
E × B drift that competes with the poloidal component of parallel streaming while staying
well below vi, since cE × B/B2 ~ (Bp/B)vi, where B = I∇ζ + ∇ζ × ∇ψ with ζ , ϑ and ψ the
toroidal angle, poloidal angle and poloidal flux variables, respectively, B = B , and ∇ψ =
€
€
€
€ €
€
€
€
RBp. For a dΦ/dr this large, the variation in potential energy over an orbit width is sufficient
to make electrostatic trapping important, even when the potential is a flux function [1-5,11].
In addition to introducing this finite orbit effect, orbit squeezing can enter [17]. T h e
competition between the E × B drift and the poloidal projection of parallel streaming makes it
necessary to retain the distinction between the surfaces of constant magnetic flux and the drift
surfaces on which the canonical angular momentum remains constant. However, to obtain
analytical results,€we must assume the inverse aspect ratio ε is small compared to unity ( ε <<
1). With this assumption the transport remains local because the trapped and barely passing
ion orbit widths are of order ε1/ 2ρ pi , and so less than the equilibrium pedestal scale length that
€
€
is allowed to be as small as ρ pi .
2 /2B,
To see how the orbits are modified we employ the magnetic moment µ = v⊥
along with the drift€approximation to the canonical angular momentum

(1)
ψ∗ ≡ψ− (Mc/Ze)R 2 v ⋅∇ζ ≈ψ − (Iv|| /Ωi ) ,
€
€ energy variable
since we consider Bp/B << 1, and the notationally convenient pseudo kinetic
and constant of the motion
€E ∗ ≡E − (Ze /M)Φ(ψ∗ ) = v 2 /2 + (Ze /M)[Φ(ψ) − Φ(ψ∗ )] ,
(2)
2
where E = v /2 + (Ze /M)Φ(ψ) is the total energy that is a constant of the motion. We also
assume the potential only depends on ψ with a quadratic dependence so that using (1) gives
€
(3)
Φ(ψ) = Φ(ψ∗ ) + (Iv|| / Ωi )Φʹ′(ψ∗ ) + (1 / 2)(Iv|| / Ωi )2 Φʹ′ʹ′ ,
€ with Φʹ′ʹ′ a constant. The utility of these variables will become clearer in the next section
€ the drift kinetic equation. In this section we obtain the relations
when they are used to rewrite
needed in subsequent sections.
We define the effective E × B velocities in the poloidal magnetic field and the orbit
squeezing factor S as
(4)
u = cIΦʹ′/B , u∗ = cIΦ∗ʹ′/B , and S = 1+ (cI2Φʹ′ʹ′/Ωi B) ,
€
with Φʹ′(ψ∗ ) = Φ∗ʹ′ and Φʹ′(ψ) = Φʹ′ . We then use Φʹ′(ψ) = Φʹ′(ψ∗ ) + (ψ − ψ∗ )Φʹ′ʹ′ to obtain the
useful and important relation
€
€
(5)
v|| + u = Sv|| + u∗ .
€ In addition, we€can use the definitions (4) and the preceding result to rewrite E ∗ as
E ∗ ≡Sv||2 / 2 + µB + u∗v|| = [(Sv|| + u∗ )2 / 2S] + µB − u∗2 /2S = [(v|| + u)2 / 2S] + µB − u∗2 /2S . (6)
2
1/2
Solving (6) for v|| gives v€
so sgn(v||+u) = sgn(Sv||+u*)
|| + u = Sv|| + u ∗ = ±(2SE ∗ + u ∗ − 2SµB)
must also be specified when ψ*, ϑ, E *, µ are used as the variables.€Unlike the more familiar
situation with ψ, ϑ, E, µ as the variables, sgn(v||) is no longer a useful coordinate since two
phase space locations in ψ*, ϑ, E*, µ can have the same sign. Phase space remains split into
trapped and passing regions defined by the preceding randicand at ϑ = π. The trapped
distribution function must be independent of sgn(Sv||+u*) at the bounce points where it
vanishes. This property is used in section 4 to define the new transit average annihilation
operation holding fixed ψ*. The preceding results are valid for arbitrary aspect ratio.
To make further progress it is necessary to introduce the inverse aspect ratio
expansion of the magnetic field by letting
B0/B = 1 - 2εsin2(ϑ/2) = 1 - ε + εcosϑ ,
(7)
where B0 is the magnetic field at ϑ = 0. Retaining inverse aspect ratio corrections through
order ε we may then write (6) as
(8)
(v|| + u) 2 /2 = (Sv|| + u∗ ) 2 /2 = W(1− ΛB/B0 ) = (1− Λ)W[1− κ 2 sin 2 (ϑ /2)] ,
2
where W, Λ and κ are adiabatic invariants to order ε defined by
2
(9)
W = S0E ∗ + 2(S0 −1)(E ∗ − µB0 ) + (3/2)u∗0
2
€
S0µB0 + 2(S0 −1)(E ∗ − µB0 ) + u∗0
κ2
(10)
Λ= 2
=
2
κ + 2ε S0E ∗ + 2(S0 −1)(E ∗ − µB0 ) + (3/2)u∗0
2 ]
€
2εΛ 2ε[S0µB0 + 2(S0 −1)(E ∗ − µB0 ) + u∗0
(11)
κ2 =
=
2
1− Λ
S0 (E ∗ − µB0 ) + (1/2)u∗0
€
with
(12)
u∗0 = cIΦ∗ʹ′/B0 , S0 = 1+ (cI2Φ/Ω0 B0 ) ,
2
and Ω0 =€ZeB0/Mc. The new variables W and Λ reduce to the familiar ones v /2 and λ =
2µB0/v2 in the limit of vanishing radial electric field ( u∗0 = 0 and S0 = 1). Equation (8) then
reduces to the €
usual result that v||2 = v2 (1 − λB / B0 ) . Equations (9) - (11) are consistent with
the expressions from [1-5] written in terms of v||0, the parallel velocity evaluated at ϑ = 0. The
trapped-passing boundary occurs at κ2 = 1 or€Λ = 1/(1+2ε), and is shifted and distorted from
the usual boundary as noted in [1,4,5]. Equations (8) - (11) are particularly convenient for
switching between ψ∗ and ψ variables. In the next section they will allow the transit averages
that follow an ion trajectory to be performed holding ψ∗ fixed, while evaluating the flux
surface averages at fixed ψ.
€
3. Drift kinetic equation
€
Using ψ∗ , ϑ, E ∗ , and µ as our independent variables for an axisymmetric tokamak
˙ ∗∂f/∂ψ∗ + ϑ˙ ∂f/∂ϑ + E˙ ∗∂f/∂E ∗ = C{f} since µ is an
results in the drift kinetic equation ∂f/∂t + ψ
adiabatic invariant. Here f = f( ψ∗ ,ϑ, E ∗ ,µ) is the gyroaveraged ion distribution function and
€
˙ ∗ = 0, allowing slow
C is €
the Fokker-Planck
collision operator for ion-ion collisions. Using ψ
˙ = (Ze/M)∂Φ/∂t, and working to high enough order to retain E × B
time dependence so that E€
∗
€
€ ϑ˙ = (v||n + v d ) ⋅ ∇ϑ with n = B/B. As a result, our drift kinetic
plus magnetic drifts, vd, gives
€
equation is simply
€
€ + (v n + v ) ⋅ ∇ϑ∂f/∂ϑ + (Ze /M)(∂Φ /∂t)∂f/∂E = C{f} .
(13)
∂f/∂t
||
d
∗
€ dependence, (13) allows us to consider the residual zonal flow
By retaining slow time
problem and neoclassical transport in the pedestal in an additive fashion. To do so we let
€
(14)
f = f∗ (ψ∗ ,E) + h(ψ∗ ,ϑ,E ∗ ,µ,t)
where
(15)
f∗ = f∗ (ψ∗ ,E) = η(ψ∗ )[M /2πTi (ψ∗ )]3 / 2 exp[−ME /Ti (ψ∗ )] ,
€
€
with both the pseudo density, η(ψ) = ni(ψ)exp[ZeΦ(ψ)/Ti(ψ)], and the ion temperature,
Ti(ψ), weakly varying functions of space in the pedestal, as required to make the entropy
production vanish [11]. Then f∗ can be Taylor expanded about ψ. Notice that the ion density
ni and the electrostatic potential are allowed to be strong functions of ψ so the density cannot
be expanded about ψ. The background potential Φ(ψ) is assumed quadratic in ψ, while the
€
fluctuating potential€ is assumed to be a small correction
that does not impact ion orbits.
Expanding (15) about the Maxwellian
€
(16)
fM = fM (ψ,E) = η(ψ)[M /2πTi (ψ)]3/ 2 exp[−ME /Ti (ψ)]
gives
⎪⎧ Iv ⎡ ∂np i Ze ∂Φ ⎛ Mv 2 5 ⎞ ∂nTi ⎤
⎪⎫
(17)
f∗ = fM ⎨1− || ⎢
+
+ ⎜
− ⎟
⎥ + ...⎬ ,
€
⎪⎩ Ωi ⎣ ∂ψ
⎪⎭
Ti ∂ψ ⎝ 2Ti 2 ⎠ ∂ψ ⎦
where dlnpi/dlnTi ~ LT/ρpi ~ B/Bp, with LT the ion temperature scale length. Notice that the
correction to fM is f* - fM + h. Then the kinetic equation (13) to the requisite order becomes
€ ∂h/∂t + (v|| + u)n ⋅ ∇ϑ∂h /∂ϑ + (Ze /M)(∂Φ /∂t)∂fM/∂E = C  {f − fM} ,
(18)
€
where we have allowed for the possibility of finite E × B effects by retaining the u term from
ϑ˙ , and use C  to denote the linearized ion-ion collision operator. For the residual zonal flow
€
calculation
collisions are neglected. For the neoclassical transport calculations considered
€ linearity of (18) means that we can add the
next, the time derivatives are dropped. The
€
neoclassical
and zonal flow contributions to the perturbed ion distribution function that are
found in the next two sections.
4. Neoclassical transport
Dropping the time derivatives and noting that for the linearized ion-ion collision
operator C  {(1,v,v2/2)fM} = 0, it is convenient to define
Iv ⎛ Mv 2 5 B2k ⎞ ∂nTi
,
(19)
H = h − fM || ⎜
− +
⎟
Ωi ⎝ 2Ti 2 2〈B2 〉 ⎠ ∂ψ
€where k is a flux function to be determined. The k term is added for later use to restore the
C  {v||fM} = 0 property when it becomes necessary to employ an approximate collision
€ explicit results. The B dependence of the k coefficient is made explicit to
operator to obtain
€
obtain a form for the flows that will be divergence-free, where 〈B2 〉 = B20 with 〈...〉 denoting a
flux surface average. Using (19) in the steady state version of (18) simplifies it to
(20)
(v|| + u)n ⋅ ∇ϑ∂h /∂ϑ = C  {H} ;
the form for the kinetic equation that is useful in both€the banana and€plateau regimes.
4.1 Plateau regime
The plateau €
regime calculation is performed in detail in [3] and easily recovered from
the preceding formalism. The procedure is to first realize that because of the singular nature
of H at v|| = 0, (20) is in the correct form to make the usual plateau replacement of C  {H} by
-νH to resolve the singularity. We then also use (5) to write the streaming operator in terms
€
of the ψ∗ , ϑ, E ∗ , and µ. Once the equation is solved for h the solution must be changed back
to the ψ, ϑ, E, and µ variables to perform the velocity space integrals holding ψ fixed (these
details of the procedure are addressed in the next subsection). The flux function k is
€ determined
€
by requiring the ions give no particle transport as required by the momentum
conservation property C  {v||fM} = 0 of the full linearized ion-ion collision operator. The
corrected result [3] is
1+ 4U 2 + 6U 4 + 12U 6
,
(21)
k = J p (U 2 ) =
1+ 2U 2 + 2U 4
€
with n ⋅ ∇ϑ = qR 0 , q the safety factor, R0 the major radius at ϑ = 0, and
cI ∂Φ
U=
.
(22)
v iB0 ∂ψ
€
€
€
When these steps are carried out, orbit squeezing does not enter and the following results are
3
obtained for the parallel ion flow V||i = n−1
i ∫ d vv||f , radial ion heat flux
〈qi⋅ ∇ψ〉 = 〈 ∫ d 3v(Mv 2 /2)fv d€⋅ ∇ψ〉 , and bootstrap current J bs :
Ip ⎡∂np i Ze ∂Φ J p (U 2 )B2 ∂nTi ⎤
(23)
n iV||i = − i ⎢
+
+
⎥ ,
MΩi ⎣ ∂ψ
Ti ∂ψ
2〈B2 〉
∂ψ ⎦
€
3/ 2
π ε2I 2p i ⎛€
Ti ⎞ ∂nTi
〈qi⋅ ∇ψ〉 = −3
L p (U 2 ) ,
(24)
⎜ ⎟
2 qR 0Ω2i ⎝ M ⎠
∂ψ
€
and
π ε2cIBv e
2 + 4Z ⎡ ∂p ( 2 + 13Z)n e ∂Te J p (U 2 )n e ∂Ti ⎤
(25)
J bs = −
+
⎢ +
⎥
€ 2 qR 0 ν e 〈B2 〉 Z(2 + 2Z) ⎣ ∂ψ 2( 2 + 4Z) ∂ψ
2Z
∂ψ ⎦
where pe = neTe, ve = (2Te/m)1/2, νe = 4(2π)1/2nee4lnΛ/(3m1/2Te3/2), and
1+ 4U 2 + 8U 4 + 4U 6 (4 + U 2 ) /3
(26)
L p (U 2 ) =
exp(−U 2 ) .
2 + 2U 4
1+
2U
€
The poloidal flow of Pfirsch-Schluter trace impurities (subscript z) is then given by
cIBp Ti ⎡∂np i ZTz ∂np z J p (U 2 ) ∂nTi ⎤
Vzp = −
−
+
(27)
⎢
⎥ .
Ze〈B2 〉 ⎣ ∂ψ
Z z Ti ∂ψ
2
∂ψ ⎦
€
The factors Jp and Lp account for the modifications of the usual plateau results [8] by finite
radial electric field effects and equal one at U = 0. The bootstrap current is evaluated by a
two-term€Laguerre expansion in combination with an adjoint method [3]. The function Jp is a
monotonically increasing function approaching an asymptote of 6U2– 3+… for large U2.
Therefore the poloidal ion and impurity flows are enhanced by the pedestal electric field for
normal Ti profiles. The function Lp increases to a maximum of 1.46 at U2 = 0.83 before going
to zero exponentially. This increase in Lp enhances the ion heat transport for normal Ti
profiles for moderate U2.
4.2 Banana regime
The banana regime calculation is somewhat more involved because of the need to deal
with the collision operator in detail while distinguishing carefully between transit (performed
at fixed ψ*) and flux surface (performed at fixed ψ) averages. The Rosenbluth potential form
of the ion-ion collision operator for collisions with a background Maxwellian is used with the
flux function k being determined by the need to conserve momentum in ion-ion collisions so
that ion particle transport does not occur. The convenient variables to employ first are the ψ,
ϑ , W and Λ variables. Only scattering normal to the trapped-passing boundary κ2 = 1 need
be retained because we assume ε << 1 and neglect order ε corrections [18]. As a result, we
need only evaluate the modified pitch angle scattering operator

1 ∂ ⎡
∂ ⎛ f ⎞⎤
(28)
C{f} =
⎢ jfM∇ v Λ ⋅ Q ⋅ ∇ v Λ
⎜ ⎟⎥ ,
€
j ∂Λ ⎣
∂Λ ⎝ fM ⎠⎦


where j = 1/∇ vW × ∇ v Λ ⋅ ∇ vϕ is the Jacobean and Q = (ν⊥ /4)(v 2 I − vv) + (ν|| /2)vv , with
ν⊥ /ν i= 3(2π)1/ 2 [erf(x) − Ψ(x)]/(2x 3 ) , ν||/ν i= 3(2π)1/ 2 Ψ(x) /(2x 3 ) , x = v/vi, erf(x) the error
function, Ψ(x)€= [erf(x) − xerf'(x)]/(2x 2 ) , νi the Braginski ion-ion collision frequency, and
€ the gyrophase ϕ giving ∇ vϕ = v⊥2 n × v . To evaluate
€
∇ vW and ∇ v Λ we use (8), ΛW from (9)
€ (1) to obtain
and (10), and ∇ vψ∗ = In /Ωi from
€
(1− ΛB/B0 )∇ vW = (B/B0 )W∇ v Λ + (v|| + u)n
(29)
€
€
and
€
€
€
W∇ v Λ + Λ∇ vW = [(S0B0 /B) − 2(B0 /B)(S0 −1)]v⊥ + 2(S0 −1)v .
€ order ε corrections the preceding give
Upon neglecting
j = BW/S0B0(v|| + u)
and€
€
€
€
(30)
(31)
∇ v Λ = −(Λ /W)(v|| + u)n + (1− ΛB/B0 )[S0 v⊥ + 2(S0 −1)v||n + ...] ≈ −(Λ /W)(v|| + u)n . (32)
The last form of ∇ v Λ is all that is needed to evaluate (28) which becomes

(v + u) ∂ ⎡
∂ ⎛ H ⎞⎤
(33)
C  {H} = ||
⎢BΛW −2 (v|| + u)n ⋅ Q ⋅ nfM
⎜ ⎟⎥ ,
B ∂Λ ⎣
∂Λ ⎝ fM ⎠⎦
where €
to reduce to the usual u = 0 result we use Λ ≈ 1 since (33) is only employed for the
trapped and barely passing. Equation (33) is easily written in terms of the ψ*, θ, W and Λ
€ by using (5), (6), and (8) - (10) so the transit averages can be performed:
variables

(Sv|| + u∗ ) ∂ ⎡
∂ ⎛ H ⎞⎤
(34)
C  {H} =
⎢BΛW −2 (Sv|| + u∗ )n ⋅ Q ⋅ nfM
⎜ ⎟⎥ ,
B
∂Λ ⎣
∂Λ ⎝ fM ⎠⎦
where ρ pi/LT ~ Bp/B << ε1/2 is assumed to neglect corrections from ∂ ψ∗/∂Λ. Notice that

combining (29) and (32) give ∇ v W ≈ (v|| + u)n and W∇ v W ⋅ ∇ vΛ ≈ −Λ(v|| + u)2 . As a result,
€
the neglected
velocity space derivatives from the full ion-ion collision operator are expected
to be small as Λ derivatives acting on H are larger than W derivatives for ε << 1.
To complete the specification of the collision operator we need to evaluate the

coefficient n ⋅ Q ⋅ n for the trapped and barely passing ( v|| + u = Sv|| + u∗ ≈ 0 ) in both sets of
variables. Using v 2 = v||2 + 2µB and the lowest order result v|| ≈ −u ≈ −u∗/S , (6) and (9) then
2 /S ≈ S µB + S u 2 , since here we can
give E ∗ ≈ µB + u∗2 /2S ≈ µB + Su 2 /2 and W ≈ S0µB0 + u∗0
0
0
0
0
1/2
2
2 /S 2 ,
€
€ gives v ≈ 2W /S − u 2 ≈ 2W /S0 − u∗0
neglect ε order corrections. Using these results
0

€
€
which enters in ν⊥ and ν|| and allows us to write n ⋅ Q ⋅ n in both sets of variables:
€
€
€
€
€

2 /2S 2 .
(35)
n ⋅ Q ⋅ n ≈ Wν⊥ /2S + (ν|| − ν⊥ )u 2 /2 ≈ Wν⊥ /2S0 + (ν|| − ν⊥ )u∗0
0
Recalling (20) and using the lowest order banana regime result ∂h /∂ϑ = 0 we see that
the transit average of the next order version of (20) with (34) inserted for C for the passing
€ allows the lowest order flux function ∂h /∂Λ to be determined from
ions
€
BC  {H}
=0 .
(36)
Sv|| + u∗
€
€
As usual, h = 0 for the trapped particles
since the transit average is over a full bounce. Notice
that the transit averages can be performed in essentially the usual manner [6-8,18]. Once
∂h /∂Λ is determined from€(36) for the passing in the ψ*, θ, W and Λ variables it is easily
rewritten in the ψ, θ, W and Λ variables and the desired moments can be formed in the
conventional manner. The flux function k is determined by the need to conserve momentum
in ion-ion collisions and is found to be
∞
3k
6 ⎡ 5
∫ o dye−y (y + 2U 2 )3/2 (yν⊥ + 2U 2 ν|| ) ⎤
2
2
(37)
≡ J b (U ) = ⎢ + U − ∞
⎥ .
7
7 ⎣ 2
∫ o dye−y (y + 2U 2 )1/2 (yν⊥ + 2U 2 ν|| ) ⎦
The details of this evaluation as well as those for the parallel ion flow velocity, radial ion heat
flux, and bootstrap current are found in [1,2] and lead to the following expressions
Ip ⎡∂np i Ze ∂Φ 7J b (U 2 )B2 ∂nTi ⎤
(38)
n iV||i = − i ⎢
+
−
⎥ ,
MΩi ⎣ ∂ψ
Ti ∂ψ
6〈B2 〉
∂ψ ⎦
〈q i ⋅ ∇ψ〉 = −1.35
and
ε νi I2 p i ∂Ti L b (U 2 )
,
MΩ2i ∂ψ
S
(39)
€
J bs = −1.46 ε
cIB (Z 2 + 2.21Z + 0.75) ⎡ ∂p
(2.07Z + 0.88)n e ∂Te 7J b (U 2 )n e ∂Ti ⎤
−
−
⎢
⎥,(40)
〈B2 〉
6Z
∂ψ ⎦
Z(Z + 2)
⎣ ∂ψ (Z 2 + 2.21Z + 0.75) ∂ψ
where
2
L b (U 2 ) = 1.53e−U ∫ o∞ dye−y (y + 2U 2 )3/2 (y + U 2 )−3/2 (y + U 2 − 5 / 2 + 7J b / 6)
€
×[yerf( y + U 2 ) + (2U 2 − y)Ψ( y + U 2 )] .
(41)
The poloidal flow of Pfirsch-Schluter trace impurities and banana regime ions is now given
by
€
cIBp Ti ⎡∂np i ZTz ∂np z 7J p (U 2 ) ∂nTi ⎤
Vzp = −
−
−
(42)
⎢
⎥ .
Ze〈B2 〉 ⎣ ∂ψ
Z z Ti ∂ψ
6
∂ψ ⎦
As in the plateau regime, the normalized factors Jb and Lb are the modifications of the
usual banana regime results [8] caused by finite radial electric field effects. The bootstrap
current €
is evaluated by inserting the Jb modification in the expressions given by Helander and
Sigmar [8]. The function Jb decreases monotonically, changing sign at U2 = 1.44. As a result,
the poloidal ion flow can differ importantly from the usual banana regime result. The
predicted change in the flow of the background ions also impacts the impurity flow as given
by (42) and is indeed observed on Alcator C-Mod [13,19]. The function Lb starts out rather
flat at small U2 before going to zero exponentially. Consequently, the radial ion heat flux is
€
reduced by the finite electric field that acts to decrease the number of trapped and barely
passing ions.
5. Zonal flow residual
Analysis of the zonal flow residual is a way of considering the response to a
turbulence generated, axisymmetric small amplitude, zonal flow potential fluctuation. This
perturbation is assumed to have rapid radial spatial variation, but no poloidal variation:
˜ (t)exp[iG(ψ)]. The turbulence is assumed to generate the change in the potential
Φ(ψ,t) = Φ
in a time long compared to an ion gyro-period, but short compared to an ion bounce time,
while not perturbing the density. The standard definition for the ion zonal flow residual is the
ratio of the long time asymptotic value of the perturbed potential to the initial perturbation. It
can be obtained by solving (18) in the collisionless limit in the absence of any neoclassical
drive (f* = fM is employed since the zonal flow and neoclassical problems are additive):
˜ /∂t)fM exp(iG) .
(43)
∂h/∂t + (v|| + u)n ⋅ ∇ϑ∂h /∂ϑ = (Ze /T)(∂Φ
We solve (43) by assuming to to lowest order ∂h/∂ϑ = 0 and then annihilating the streaming
term to next order by employing the transit average along the actual trajectory over a full
€ transit. Using the definition
poloidal


∫ dϑA /(v|| + u)n ⋅∇ϑ ∫ dϑA /(Sv|| + u∗ )n ⋅∇ϑ
A=
=
,
(44)


∫ dϑ(v|| + u) / n ⋅∇ϑ
∫ dϑ(Sv|| + u∗ ) / n ⋅∇ϑ
and periodicity in ϑ , we obtain the solution
€
˜ fM exp(iG) ,
(45)
h = (Ze /Ti )Φ
€
where the transit average of the eikonal G must be performed holding ψ∗ , W and Λ fixed.
€
ZeΦ˜ /Ti << 1 and forming the flux surface averaged perturbed ion density
Assuming
˜ fM /Ti )]〉 gives
n˜ i = 〈 ∫ d 3v[hexp(−iG)€
− (ZeΦ
˜ /Ti )〈 ∫ d 3vfM [exp(iG)exp(−iG)
€ −1]〉 .
(46)
n˜ i = (ZeΦ
If gyroradius, as well as drift, departure polarization effects are retained then the preceding
€
expression becomes
˜ /Ti )〈 ∫ d 3vfM [J 0 exp(iG)J 0 exp(−iG) −1]〉 ,
€
(47)
n˜ i = (ZeΦ
where the Bessel function J 0 = J 0 (k r v⊥ /Ωi ) has k r = ∇G = RBpGʹ′. For a turbulent change at
t = 0 in a time much less the transit time (so the ions have gyrated many times, but not yet
˜ (t = 0) , the initial density n˜ i of (47) is simply
€
drifted significantly)
to Φ
˜ /Ti )〈 ∫ d 3vfM [J 0 J 0 −1]〉 .
€
(48)
n˜ i (t = 0) = (Ze€
Φ
˜ leaves the ion density unchanged and wait for Φ˜ to settle to its
If we assume this change in Φ
˜ (t → ∞) , while keeping n˜ i (t → ∞) = n˜ i (t = 0) , then we can form the
time asymptotic value Φ
€
€
€ (48) to obtain
ratio of (47) and

〈 ∫ d 3vfM [J 0 J 0 −1]〉
€ Φ(t → ∞) =
€.
(49)
 = 0) 〈 ∫ d 3vf [J exp(iG)J exp(−iG) −1]〉
Φ(t
M
0
0
€
€
For small kr and u (49) reduces to the form of Rosenbluth and Hinton [9,10].
€
€
€
To extend their result to finite u we need to Taylor expand G in a way that the lowest
order term depends only on ψ∗ , W, Λ, and constants, and the next corrections are small in
ε so that exponentials can be expanded in (49). We start by writing
(50)
ψ = ψ∗ − (Iu∗ /SΩi ) + [I(Sv|| + u∗ ) /SΩi ] ,
so the last term is a€ ε correction for the trapped and barely passing ions - the only ones of
interest for our evaluation. Taking account of the poloidal variation of B in u and S as well as
€ ion orbit results of section 2 to retain order ε terms, we can write
Ωi and using the
€ = Ψ + [I(Sv + u ) /SΩ ] + [4εIu /S2Ω κ 2 ][1− κ 2 sin 2 (ϑ /2)] + ... ,
(51)
ψ
∗
||
∗
i
∗0 0 0i
2
2
where Ψ∗ ≡ ψ∗ − (Iu∗0 /S0Ω0i )[1− 4ε /S0 κ ] . We then Taylor expand G about Ψ∗ to obtain
€Q − L + ... ,
(52)
G(ψ) = G∗ + P + ... = G∗ +
€
where G∗ = G(Ψ∗ ) , Gʹ′∗ = Gʹ′(Ψ∗ ) , P = Q - L, Q = [I(Sv|| + u∗ ) /SΩi ]Gʹ′∗ , and
€ L = [4εIu∗0 /S20Ω0iκ 2 ][1− κ 2 sin 2 (ϑ /2)]G∗ʹ′ . We neglect Gʹ′ʹ′ and€ smaller terms when
performing the€ Taylor expansion because the eikonal has the usual property that
€ ∇∇G ~ k€
small kr then gives
r /a . Expanding (49) for €
 → ∞)
Φ(t
1€
=
,
(53)
 = 0) 1+ ℜ
Φ(t
with
€
2 2 −1
ℜ = 2n −1
∫ d 3vfM [i(P − P) + (P 2 − 2PP + P 2 ) / 2]
i 〈k r ρ i 〉
(54)
to the requisite order. Noting that L ~ ε Q, then only linear terms in L need be retained in
(54) along with linear and quadratic terms in Q. The evaluation then proceeds as in
Landreman and Catto [5], where the full details are presented, to find
⎡ Γ(U 2 ) Λ(U,S) ⎤
€
(55)
ℜ = ℜ RH ⎢
+i
⎥ ,
〈k r ρ pi 〉 ⎦
⎣ S
where ℜ RH = 1.6q 2 / ε is the Rosenbluth and Hinton result,
∞
€
and
Γ(U 2 ) = (4/3 π )exp(−U 2 ) ∫ dy(y + 2U 2 )3/2 exp(−y)
0
€
∞
⎡
⎤
Λ(U,S) = 2S−1/2 U ⎢SΓ(U 2 ) + 4π−1/2 (1 − S)exp(−U 2 ) ∫ dy(y + 2U 2 )1/2 exp(−y)⎥ .
⎣
⎦
0
(54)
(54)
The exponential decay of Γ and Λ for large U is due to the shift of the trapped region to the
tail of the distribution function, orbit squeezing effects only enter algebraically in Λ, and the
 introduced by u.
imaginary term is a spatial phase shift in Φ
6. Discussion
We have presented a streamlined evaluation of the ion orbits and a generalized kinetic
treatment that allows us to recover all the pedestal results obtained to date for neoclassical ion
flow and heat flux and the bootstrap current in the banana [1,2] and plateau [3] regimes, and
for the zonal flow residual in the collisionless limit [4,5]. The techniques we employ clearly
distinguish between trajectory and flux surface averages and unify recent evaluations of
pedestal phenomena. These modifications include the banana regime decrease and plateau
regime alterations in the ion heat diffusivity, the reduction and possible reversal of the
poloidal flow in the banana regime and its augmentation in the plateau regime, the increase in
the bootstrap current, and the enhancement of the residual zonal flow regulation of
turbulence.
Acknowledgements: Work supported by U. S. Department of Energy grants at DE-FG0291ER-54109 at MIT and DE-AC52-06NA-25396 at LANL, and the European Communities
under Association Contract between EURATOM and Vetenskapsradet. PJC is grateful for the
hospitality of the Isaac Newton Institute for Mathematical Sciences in Cambridge, England
while preparing portions of this article.
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